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A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes.
For example, 3 is a Mersenne prime as it is a prime number and is expressible as 2 2 − 1. [1] [2] The exponents p corresponding to Mersenne primes must themselves be prime, although the vast majority of primes p do not lead to Mersenne primes—for example, 2 11 − 1 = 2047 = 23 × 89. [3]
Prime ideals, which generalize prime elements in the sense that the principal ideal generated by a prime element is a prime ideal, are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry.
The table below lists the largest currently known prime numbers and probable primes (PRPs) as tracked by the PrimePages and by Henri & Renaud Lifchitz's PRP Records. Numbers with more than 2,000,000 digits are shown.
A prime sieve works by creating a list of all integers up to a desired limit and progressively removing composite numbers (which it directly generates) until only primes are left. This is the most efficient way to obtain a large range of primes; however, to find individual primes, direct primality tests are more efficient [ citation needed ] .
Because the set of primes is a computably enumerable set, by Matiyasevich's theorem, it can be obtained from a system of Diophantine equations. Jones et al. (1976) found an explicit set of 14 Diophantine equations in 26 variables, such that a given number k + 2 is prime if and only if that system has a solution in nonnegative integers: [7]
The current prime rate is 5.50%, up from 4.75% in June. It went into effect July 28, 2022. This is the fourth time in 2022 that the Federal Reserve has increased the prime rate.
For example, the AP-3 {3, 7, 11} does not qualify, because 5 is also a prime. For an integer k ≥ 3, a CPAP-k is k consecutive primes in arithmetic progression. It is conjectured there are arbitrarily long CPAP's. This would imply infinitely many CPAP-k for all k. The middle prime in a CPAP-3 is called a balanced prime.